Just a word of advice to add in. If tires are yucky, that is a $50.00 expenditure for decent ones to replace. Tires and wheels make the difference with older bikes.
You might want to increase your budget by the $50 for "tricking out" and see if $175-200 gets you something ready to go.
I notice there is a vintage 60cm Bottechia for sale on there.
And a larger Trek
https://chicago.craigslist.org/chc/b...884862212.html
And my choice for your height/inseam - 58cm, brifters. $175
https://chicago.craigslist.org/chc/b...898449210.html
Lots of choices down here in Charleston...